Aircraft: SMALL TRANSPORT
Phase of flight: Final Approach
Anomaly: Aircraft Equipment Problem Less Severe; Deviation - Altitude Excursion From Assigned Altitude; Deviation - Track / Heading All Types; Deviation / Discrepancy - Procedural Clearance; Deviation / Discrepancy - Procedural Published Material / Policy

I was PIC on a flight going to OTH. When I first checked weather in OTH it was at 8 miles vis.; but it started deteriorating rapidly. But due to the nature of the coastal weather; I had planned for this and took on extra fuel. Our original destination had been ZZZ1; but we had to file for OTH because ZZZ1 was below minimums. Seattle Center cleared me for the ILS Runway 4 via the 16 DME Arc from LUPCI. Things were going smoothly until I realized as I was approaching LUPCI for the approach that I hadn't heard from Seattle Center for a while and I was still high. I realized that I was still on CTAF to activate the approach lights and hadn't switched back. I contacted Seattle Center and; frustrated; he cleared me down to 2400 feet for the approach. I was able to get down to 2400 feet and stabilized in order to make the ILS intercept.The DME ARC and ILS Intercept went fine; and I continued down to minimums. My autopilot clicked off; but I reset it through the mode control panel with no problem. With no runway environment in sight; I executed the published missed approach INITIALLY; with a climb to 600 feet and then a climbing left turn back to EMIRE LOM. It was here that I got confused on the Controllers 'state your intentions' remark that I started on the wrong path. I don't know why I thought that I could do this; but somehow I thought that I could re-intercept the 167 radial off of OTH VOR and start another DME Arc to have a better amount of time for the weather to clear. Again; looking back; I knew that I was in a non-radar environment; so I don't know why I thought that I could be off the published missed approach but I was. The Controller expressed concern and instructed me to climb immediately to 4000 feet because of terrain. I wasn't indicating any terrain conflict on my TCAS or Garmin 650/750 combo; but the Controller advised me of terrain in that area. The Controller advised me to proceed back to EMIRE and execute the published missed approach the way I should have done it initially; with the procedure turn; etc. I was a little turned around by this point; but got back on course to EMIRE. This time; because of my obvious newfound stupidity and ineptitude; he read me the entire missed approach instructions verbally so I wouldn't screw up another missed approach.At one point; my autopilot clicked off again; but I don't remember how high I got; 200 feet above charted I think; but I thought that it was maybe something I did or the surprising amount of turbulence for the low coastal visibility. This time; I hand flew the procedure turn inbound and the ILS 4 and broke out just above minimums with a 12 knot crosswind. I landed uneventfully and cancelled my flight plan. I thanked the Controller and I think that I apologized; but if I didn't I should have. From OTH; we flew uneventfully. On a side note; on our return flight; my autopilot clicked off again in smooth air; so I wrote it upon arrival. While I haven't done very many DME Arcs in my career; my DME Arc wasn't the problem; it was my home made missed approach procedure. Since then; I have been brushing up on my non-radar procedures; especially missed approaches to LOM's with a procedure turn outbound. The charter pilot did not fly the complete missed approach procedure at OTH and was corrected by ATC; even though it was a non-radar environment.